Universal nature of rapid evolution of conservative gravity and turbidity currents perturbed from their self-similar state

The results of six highly resolved direct and large eddy simulations of gravity and conservative turbidity currents are presented to illustrate the point that the rapidly varying evolution of these currents follow a universal cyclic sequence of four states. The demarcation between these four states is determined by the bulk Richardson number and the acceleration/deceleration of the flow. We describe in detail the identification process of these states, together with the associated three-dimensional structure of the current. The exact depth-averaged momentum balance is computed and used to explain the intricate details of the nonmonotonic rapid evolution of the current between the different states. Finally, the balance of turbulent kinetic energy and concentration flux are computed to explain how and why the current evolves through the cyclic sequence of states. We also explore the mechanism by which the current can exit the cyclic sequence and slowly evolve towards self-similar supercritical or subcritical states.

Figure 1

(a), (b) bulk Richardson number $\mathrm{Ri}$ and the local minimum value of the gradient Richardson number ${\mathrm{Ri}}_g$ within the interface layer for (a) case 1 and (b) case 2. Colors denote the non-normal regime at each streamwise location. (c) Spanwise and time averaged concentration field for case 2. Also shown are current thickness $h\left(x\right)$, upper edge of the current defined by the location of zero streamwise velocity $h_u\left(x\right)$, and the location of the streamwise velocity maximum $z_{um}\left(x\right)$. Wall normal axis is stretched for better visualization.

Figure 2

Scaled velocity and concentration profiles at different downstream locations for (a) case 1 and (b) case 2. Experimental and field data points from $\square$ [19], $◃$ [35], $◯$ [36], $\mathrm{\Delta }$ [37], and $\diamond$ [12]. Blue and red symbols correspond to velocity and concentration profiles, respectively. The dashed blue line corresponds to $z_{um}\left(x\right)$.

Figure 3

Three-dimensional structure of case 1. Turbulent structures in the density current are captured by an isosurface of swirling strength (${\lambda }_{ci}=4$ at interface and ${\lambda }_{ci}=10$ at near-bed layer) and colored by wall normal location $z$, together with bottom and top isosurfaces where TKE production is zero (light blue and pink surfaces). Also shown are contours of spanwise vorticity ${\mathrm{\Omega }}_y$ at $y=1.95$ (red-blue color scale). Moreover, contours of zero TKE production at $y=1.95$ are shown in light yellow. Colors of bottom labels denote the rapidly varying state at each streamwise location.

Figure 4

Three-dimensional structure of case 2. Turbulent structures in the density current are captured by an isosurface of swirling strength ${\lambda }_{ci}=4$ and colored by wall normal location $z$. Colors denote the rapidly varying state at each streamwise location.

Figure 5

Scaled depth-averaged momentum balance as a function of downstream location $x$ for (a) case 1 and (b) case 2. Scaled depth-average TKE balance as a function of downstream location $x$, for (c) case 1 and (d) case 2. Dashed lines show the depth-average terms integrated up until the velocity maximum (near-bed balance). Balance of scaled concentration flux as a function of downstream location. (e) Near-bed balance for case 1. (f) Near-bed balance for case 2.

Figure 6

Bulk Richardson number $\mathrm{Ri}$ (blue line) together with the local minimum of gradient Richardson number ${\mathrm{Ri}}_g$ at the interface layer (red line) for (a) case 3, (b) case 4, (c) case 5, and (d) case 6. Colors denote the non-normal regime at each streamwise location.

Figure 7

Schematic representation of the near-bed and interfacial turbulence for (a) self-similar supercritical currents, (b) self-similar subcritical currents, and (c) rapidly varying states. In blue dashed line: $z_{um}$; in orange dashed line: vertical location where $\overline{u^{\prime }{w}^{\prime }}=0$. We note that in (c) the D-Sub and A-Sub states have been stretched in the streamwise direction for ease of visualization.

Figure 8

Partitioning of concentration flux between the near-bed (blue line) and the interface (red line) layers for (a) case 1 and (b) case 2.

Figure 9

Schematic momentum balance for (a) D-Super, (b) D-Sub, (c) A-Sub, (d) A-Super, (e) slow variation to SS-Sub, and (f) slow variation to SS-Super. The shown momentum balance terms are C, concentration, Ac, depth averaged acceleration, Pr, streamwise pressure gradient force, Sh, basal drag, En, ambient fluid entrainment, and Sp, depth averaged streamwise change of Reynolds stress.